Good video, cheap little HF hydraulic press will do wonders and you can service your pinion bearings with it too. I do the same thing with wheel bearings using the old races to drive new bearings home.
For install Bearing race never use hammer possible damage.
Contact when driving the bearing on is only made on the race, not the cage, and it is done with a sacrificial old bearing race in between. There's nothing wrong with doing it this way and you will not damage the bearing.
I like your use of the old bearing race to start the new one safely!👍